In 1845, James Prescott Joule published his celebrated experiment that established the mechanical equivalent of heat.1 He connected the falling of a weight to the rotation of paddles inside a water container, and measured the increase of temperature of the stirred water. Thereby he found a direct relation between the loss of potential energy by the descending weight and the heat transferred to the water. This result encouraged the contemplation of heat as a form of energy against the opposing theory of the caloric, a supposedly indestructible fluid flowing between bodies that exchange heat.2 In due time, after this and other similar experiments, the conservation of energy (including heat transfer) was recognized and sanctioned as the first law of thermodynamics.

In a sense, it was fortunate for the advancement of science that Joule used the gravitational energy of a weight as the mechanical source for agitating the water. Had he used the elastic energy of a stretched rubber band, he might have arrived at a completely different conclusion. For if he had used a rubber band—a latter-day move, since vulcanization had been discovered merely 6 years before—he might have noticed that the relaxing rubber took from the environment the same amount of heat released into the water. Just as a proponent of the caloric theory would have predicted.

This note aims to explain why this is so, and how is it possible that the purported elastic energy of stretched rubber (or other similar materials) can disappear leaving no trace of being converted into heat or other forms of energy.

This communication was originally motivated by a challenging problem found in a thermodynamics textbook by Harold Morowitz:3 a metallic spring is compressed, tied in the compressed state with a rope, and submerged into strong acid that dissolves the spring (but not the rope). What happened to the elastic energy of the compressed spring?

When mechanical energy disappears from a macroscopic body, one is led to suspect that it might have been converted into heat, that is, to the kinetic and potential energies at the atomic and molecular level that exchange freely as a result of thermal motion. After all, this is what we observe in the real world repeatedly, when movement ceases due to friction. However, as the chemical reaction of metal with acid does itself release heat, we have to postulate that (and explain why) the reaction of acid acting on a compressed spring releases more heat than on a relaxed spring. Furthermore, the excess heat must match exactly the elastic energy of the spring, so that total energy is conserved.

To understand this we should realize that disturbing a metal spring from its relaxed position—by shortening or extending it—means to force interatomic distance out of its potential energy minimum (Fig. 1). Thus, the work done at compressing the spring is stored at the atomic level as increased potential energy of the metallic bonds. Because the heat released in a chemical reaction is just the difference between products and reactants regarding the potential energy of their bonds, the excess potential energy stored in the compressed spring (compared to the relaxed one) will show up as an excess heat during the reaction.

Fig. 1.

The potential energy (V) of a bond as a function of the interatomic distance (x). The particular shape of the curve results from a combination of long-range attraction and short-range repulsion. The potential energy of the infinitely separated atoms is arbitrarily chosen to be zero. The minimum of the potential energy is located at the average interatomic distance (x0) of the unperturbed bond. A displacement to a shorter average distance (xC) due to compression is accompanied by an increase in potential energy (ΔV).

Fig. 1.

The potential energy (V) of a bond as a function of the interatomic distance (x). The particular shape of the curve results from a combination of long-range attraction and short-range repulsion. The potential energy of the infinitely separated atoms is arbitrarily chosen to be zero. The minimum of the potential energy is located at the average interatomic distance (x0) of the unperturbed bond. A displacement to a shorter average distance (xC) due to compression is accompanied by an increase in potential energy (ΔV).

Close modal

Moreover, the excess potential energy of the bonds is the cause of the restoring force (and, hence, of the elastic properties) of the spring. Because of the existence of potential energy (V), a derived (conservative) force will appear

(1)

that will be non-zero as long as the bond length (x) does not reach the potential minimum where dV/dx=0 (at x0 in Fig. 1). The elastic energy (E) will be the integral of this force along the displacement

(2)

which is exactly the excess heat released by the compressed spring upon chemical reaction with acid. Thus, in this case, the elastic energy of the spring is undoubtedly conserved.

Let's consider now a different kind of spring. Imagine a long linear polymer in a viscous liquid medium but with one end fixed to a rigid wall (Fig. 2). The polymer might be thought of as a chain of a fixed number of straight links joined with enough rotational freedom as to permit each new link to extend freely in three-dimensional space in a direction that is not correlated with that of the previous link. If the polymer does not establish interactions with itself (i.e., it is a non-sticky or non-interacting polymer), its conformation will be continuously changing due to thermal agitation and, in particular, its head to tail (or end-to-end) length, will be dictated by probability. Therefore, it is less likely for the polymer to be found in an extended configuration (as in Fig. 2(a)) than in a more retracted one (having a shorter end-to-end distance, as in Fig. 2(b)). This is because an extended (quasi-straight) polymer is compatible only with a reduced number of alternative configurations of the bond directions, while the curly macromolecule allows many more, different configurations within the same end-to-end length. As a result, there will be a distribution of polymer conformations with probability density peaking around a relatively short end-to-end distance. A simplified model of polymer conformation (termed the “random flight” model) predicts that the most likely end-to-end distance (rm) will be (see Subsection 1 of the  Appendix)4 

(3)

where n is the number of links of the polymer and l is the link length.

Fig. 2.

A linear polymer in an extended (a) or coiled (b) conformation. In the figure, one end of the polymer is kept fixed at a rigid wall while the other end is connected to a cargo ball, the whole being submerged in a viscous liquid medium. The polymer is assumed to be non-sticky (i.e., non-selfinteracting) and its conformation is ruled by mere probability. Because of the larger number of random configurations that are compatible with the depicted end-to-end distance, the coiled conformation is much more likely than the extended one. The probabilistic drive to recoil the extended polymer will result in a force (F) that will be dragging the cargo ball, thereby performing work.

Fig. 2.

A linear polymer in an extended (a) or coiled (b) conformation. In the figure, one end of the polymer is kept fixed at a rigid wall while the other end is connected to a cargo ball, the whole being submerged in a viscous liquid medium. The polymer is assumed to be non-sticky (i.e., non-selfinteracting) and its conformation is ruled by mere probability. Because of the larger number of random configurations that are compatible with the depicted end-to-end distance, the coiled conformation is much more likely than the extended one. The probabilistic drive to recoil the extended polymer will result in a force (F) that will be dragging the cargo ball, thereby performing work.

Close modal

Therefore, if we stretch out the polymer to an end-to-end distance that is much longer than rm, the incessant bumping of molecules due to thermal motion will drive the retraction of the polymer to a shorter length. Indeed, if we attach a massive particle at the free end of the extended polymer, the particle will be dragged near the other end by the polymer recoil (Fig. 2), thereby performing work (due to friction opposing the directional displacement of the particle in a viscous medium) at the expense of thermal energy. This work will be carried out by a force that does not result from a potential energy gradient but from an initial condition—the protracted position—which has low entropy (i.e., a reduced number of conformational configurations). Hence, it is called an entropic force.5 

If the process takes place at a constant temperature (T), the decrease of the Helmholtz free energy (A=UTS) dictates the available energy for work.6 Therefore, the force acting on polymer retraction can be calculated by differentiating (A) with respect to the end-to-end distance (r)

(4)

Here, the term dU/dr is null because the polymer is non-interacting and, consequently, the internal energy (U) does not vary with r. Therefore, the directional movement is driven solely by an increase of entropy (S) with displacement, which illustrates the provenance of a typical entropic force.7 

Because entropy is just a logarithmic measure of the number of configurations compatible with r, the probability density for the different end-to-end distances of the polymer (obtained from the random flight model) can be used to estimate dS/dr, thereby arriving to the result (see Subsection 2 of the  Appendix)8 

(5)

where kB is Boltzmann constant. From this relation, it follows that the force will be negative for r>rm, positive for r<rm, and will vanish at r=rm. Moreover, for sufficiently large r, the force will approximate Hooke's law with an elastic constant of 3kBT/(nl2).

As explained above, a stretched non-interacting polymer behaves as a molecular spring. It turns out that many elastic materials of natural or synthetic origin embody a cross-linked network of such microscopic springs (Fig. 3). A case in point is rubber.9 Natural rubber is a linear polymer of isoprene units obtained from the latex of certain plants (mostly from the rubber tree Hevea brasiliensis). The isoprene polymers are cross-linked through sulfur atoms in a chemical process called vulcanization in order to obtain a more resistant material. This is the common rubber that has found multiple uses in our industrial society. Due to the huge difference in length between the coiled and the unfolded polymer chains, rubber offers high extensibility storing a large work potential.

Fig. 3.

Conformational states of an elastic polymer such as rubber, resilin, or elastin. The elastic material consists of a network of linear polymer chains (thick lines) joined by crosslinks (thin double lines). The polymer molecules change from a coiled to an extended conformation through stretching of the material. In the case of elastin, the polymer chains can be cut to pieces by a treatment with the enzyme elastase, thereby losing its elastic properties.

Fig. 3.

Conformational states of an elastic polymer such as rubber, resilin, or elastin. The elastic material consists of a network of linear polymer chains (thick lines) joined by crosslinks (thin double lines). The polymer molecules change from a coiled to an extended conformation through stretching of the material. In the case of elastin, the polymer chains can be cut to pieces by a treatment with the enzyme elastase, thereby losing its elastic properties.

Close modal

Animals have also resorted to the same entropic strategy to attain elastic body parts with definite purposes. This function is usually entrusted to polypeptides (i.e., linear polymers of amino acids) with a disordered conformation. These polypeptides are collectively known as elastomeric proteins.10 For example, fleas and grasshoppers store muscular work from their rear legs in a stretched ligament made mostly from an elastomeric protein called resilin. The sudden release of the stored elastic energy allows the amazing jumps that are distinctive of these insects. Resilin is made of disordered polypeptides crosslinked through tyrosine bridges, its elasticity resulting from entropic forces as in rubber. Moreover, regarding elastic recovery and resilience, resilin is considered superior to rubber and has attracted considerable interest as a potential substitute for industrial uses requiring enhanced elastic performance.

Mammals and, in particular, humans also possess elastomeric proteins. Elastin, which is also constituted by cross-linked disordered polypeptides, provides elasticity to most human tissues.11 As habitually happens to proteins, elastin is being simultaneously synthesized and degraded in order to adapt its amount to changing conditions. Degradation of elastin is the work of specialized proteolytic activities such as elastase, which cuts the polypeptide chains into pieces (Fig. 3). Elastin accumulates in specific ligament bundles and in the skin. With age, the amount of elastin is reduced by degradation, leading to elasticity loss and wrinkles that are characteristic of elderly skin. Understandably, this process has been a subject of active research and of permanent interest for the cosmetics industry.

Consider now a variation of the problem posed by Morowitz: a bundle of elastin is kept stretched between two glass rods and submerged into a solution of elastase until all restoring force is lost. What happened to the elastic energy of the stretched elastin?

In this case, elastase catalyzes the hydrolysis of the peptide bonds of elastin in a process that may also exchange heat. However, the potential for doing elastic work of the stretched elastin resided solely in the extended conformation of the polypeptides. As long as there is no internal energy difference between the extended and relaxed conformations of elastin, there will be also no difference regarding the heat exchanged by the elastase digestion of elastin in either conformation. Thus, the elastic energy of the elastin bundle will disappear without a trace.

So, where does the elastic energy go? First of all, of what stored energy are we talking about? Surely work has to be performed in order to stretch the elastin or the rubber band. However, since the internal energy of the polymer does not change with extension, it cannot hold energy and the work done must be immediately released as heat. This is a well-known property of rubber: upon stretching, heat is simultaneously released in an equal amount to the work done.12 This crucial previous step does not only ensure that energy is actually conserved but also evidences that the stretched band did not store the elastic energy. As in a classic illusionist's trick, the energy was already not there before announcing that it will vanish.

In contrast with the metallic spring, elastic materials such as rubber or elastin do not store energy at the microscopic level when extended. What they store is a low entropy conformation13 that endows the material with the potential to perform work by recruiting thermal energy. Indeed, if we compel the stretched rubber to relax against a resisting force (thereby performing work), the rubber will cool down or, under thermostatic conditions, will take as much heat from the environment as work has been done. On the other hand, if the polymer is allowed to recoil freely without opposition or is subjected to chemical degradation, thereby losing the extended conformation that sustains the low entropy condition, no work will be done, and the potential for performing it will be lost without further energy exchange.

In elementary mechanics, potential energy is customarily assigned to physical settings that are able to perform work, and the amount of potential energy is measured as the maximum work obtainable from this configuration. Consequently, both the metallic spring and the rubber (or elastin) band are supposed to hold potential energy when stretched. In both cases, the potential energy (called elastic energy) is similarly calculated

(6)

where x is the displacement from the relaxed position and k is the elastic constant of the material.

However, we should admit that these forms of elastic energy do differ substantially. The elastic energy of the spring can be traced down to other forms of potential energy at the microscopic level. Hence, it can be said to be stored inside matter and will be always released somehow whenever the material is destroyed. In contrast, the elastic energy of rubber or elastin cannot be traced to another form of stored energy at the microscopic level and will disappear whenever the material is disassembled without performing work. This does not contradict energy conservation because, as mentioned above, the work done on stretching the elastic band was already returned as heat before the band is relaxed. The stretched band does not contain extra energy but a low entropy configuration that confers the faculty of engaging thermal energy to perform work. It is this faculty (not the energy) that is lost when the elastin is digested by elastase.

This has been always clear from a thermodynamic point of view. The potential for doing work in a thermostatic environment is given by the decrease of the Helmholtz free energy during the process. Considering a differential decrease

(7)

Therefore, work can be performed from a decrease of the internal energy (dU) or from an increase in entropy (TdS). In the first case, acting forces will be conservative and the internal energy decrease will show up at the end in one form (work) or another (heat). In the second case, forces will be entropic and the entropy increase will drive the thermal energy of the system to perform the work (subsequently forcing heat to enter from the environment to reestablish the temperature). Entropy will increase all the same if no work is done during relaxation, but then the potential for doing it will simply disappear. The thermal energy that could have been used to perform work will stay in the system, but now it will be useless for work because the entropic drive is lost. That is, free energy is converted into energy bound to the system because the entropy has been allowed to increase without performing a compensatory work.

To conclude, can mechanical energy actually vanish? The answer will be “yes” if you (wrongly) believe that there is a true potential energy stored in the stretched rubber or elastin band, as in the case of the metallic spring. If so, the presumed energy will certainly disappear without a trace when elastin is digested. On the contrary, the answer will be “no” if you (correctly) realize that rubber or elastin do not store the energy inverted in stretching but only free energy as a potential of retrieving thermal energy to do work. This potential will vanish if the material is destroyed, but this is not extraordinary because, as stated by the second law, free energy is not conserved.

1. The most likely head to tail distance in a non-sticky flexible linear polymer

Consider a linear (i.e., non-ramified) polymer made up of n straight links of length l that are freely jointed. Assuming that there are no internal interactions and that the polymer will not be unduly extended, each link might be thought as randomly oriented in space without correlation with the orientation of any other (and, in particular, with the neighboring) links. For an extremely thin polymer (so that overlappings can be neglected), the head to tail distance (r) is expected to be distributed as a 3D-random walker14 executing n steps of length l, each step having an average one-dimensional projection on an arbitrary axis equal to δ=l/3 (Fig. 4).Therefore, the distribution of r (around the polymer head located at the origin) will be a normal with variance σ2=nδ2=nl23, and with a weighting factor of 4πr2 due to the fact that r is computed as distance modulus (i.e., disregarding orientation). Thus, the probability distribution will be

(A1)

where B stands for some normalization factor. This is the random flight model of polymer conformation. The most likely end-to-end distance will be the one for which f(r) attains a maximum. Thus, making b=32nl2, we look for the extrema

(A2)

and we find a minimum for r=0 and a maximum for r=1b=l2n3, which is the result given as Eq. (3) in the main text.

Fig. 4.

The random flight polymer model. One end of the polymer is located at the origin and the backbone chain (composed of a high number of freely jointed straight links) follows the path of a three-dimensional random walker. The broken line arrow indicates the polymer end-to end distance (r). The link length (l) and its projection on the x- axis (lx) are shown in a zoom of the polymer chain. The average value of lx for all links is δ, which statistically approaches l/√3 as the number of links (n) increases.

Fig. 4.

The random flight polymer model. One end of the polymer is located at the origin and the backbone chain (composed of a high number of freely jointed straight links) follows the path of a three-dimensional random walker. The broken line arrow indicates the polymer end-to end distance (r). The link length (l) and its projection on the x- axis (lx) are shown in a zoom of the polymer chain. The average value of lx for all links is δ, which statistically approaches l/√3 as the number of links (n) increases.

Close modal
2. The retracting force of a random flight polymer

The number of polymer configurations [W(r)] for a given end-to-end distance will be proportional to the probability distribution function [f(r)] found in Subsection 1 of the  Appendix. Since entropy is a logarithmic measure of the number of configurations multiplied by the Boltzmann constant (S=kBlnW), we may write the entropy of the polymer as

(A3)

where the constant C lumps together the terms derived from the proportionality constants as well as all other contributions to entropy that do not depend on the end-to-end distance.

Because we are assuming that the retracting force is purely entropic, it can be calculated as

(A4)

which is stated as Eq. (5) in the main text.

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